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On a class of \(\delta\)-supplemented modules. (English) Zbl 1310.16004

Let \(M\) be a unitary right \(R\)-module over a ring \(R\) with identity and let \(m\in M\). This paper marries the concepts of \(\delta\)-supplemented modules [M. Tamer Koşan, Algebra Colloq. 14, No. 1, 53-60 (2007; Zbl 1111.16004)] and principally \(\oplus\)-supplemented modules [S. H. Mohamed and B. J. Müller, Continuous and discrete modules. Cambridge: Cambridge University Press (1990; Zbl 0701.16001)] as follows: A submodule \(L\) of \(M\) is said to be a principally \(\oplus\)-\(\delta\)-supplement of \(mR\) in \(M\) if \(M=mR+L\) and \(mR\cap L\) is \(\delta\)-small in \(L\) (i.e. whenever \(L=(mR\cap L)+K\) and \(L/K\) is singular, we have \(L=K\)). The module \(M\) is called principally \(\oplus\)-\(\delta\)-supplemented if every cyclic submodule of \(M\) has a principally \(\oplus\)-\(\delta\)-supplement in \(M\). Relations between principally \(\oplus\)-\(\delta\)-supplemented modules and other classes of modules, such as lifting modules, supplemented modules, principally \(\delta\)-lifting modules and principally \(\oplus\)-supplemented modules are investigated.
Conditions are found for the class of \(\oplus\)-\(\delta\)-supplemented modules to be closed under homomorphic images and direct summands. Results are obtained in the case of distributive modules, weak-duo modules, modules with \(D_3\) and duo modules. For a module \(M\), \(\text{Rad}_\delta(M)\) is defined as \(\text{Rad}_\delta(M)=\sum\{L\mid L\text{ is }\delta\text{-small in }M\}\). Relations between \(\oplus\)-\(\delta\)-supplemented modules and other concepts are studied for the case when \(\text{Rad}_\delta(M)=0\). A ring is defined to be a right \(\delta\)-V-ring if \(\text{Rad}_\delta(M)=0\) for all right \(R\)-modules \(M\) and the different classes of supplemented modules are compared in the case of modules over these rings. Principally \(\delta\)-semiperfect modules are also defined and studied in relation to different types of supplemented modules.

MSC:

16D80 Other classes of modules and ideals in associative algebras
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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